UV Mapping before Applying Modifiers
Thanks to Yohello 1, I was able to solve the problem. There is no need for a separate round of UV unwrapping, as long as you haven't applied the modifiers.

This means, that if an object has an array, and you UV unwrap before applying the array modifier, then after you apply the array modifier, or any modifier, it will also use the new meshes on the same UV map as the first it was based on.